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For cloud systems a set of stochastic models taking account of system context actualization is proposed. These models are based on queueing theory and extend its applications by studying the multichannel systems with warm-up and hyperexponential phasetype approximation of non-Markovian processes. Transition diagrams and matrices for the microstates of a sample cloud system are described and a scheme for computing the stationary probability distributions for requests number and waiting time is developed. The paper discusses the received computation results of the proposed modeling approach and their application to assessing the performance of the cloud systems.